Spherical Potential Well

The idealized infinite-walled one-dimensional and three-dimensional square-well potentials can be solved by the Schrodinger equation to give quantized energy levels. For the case of a nucleus, a useful idealization is an infinite-walled spherical potential. That is, we model the nucleus with a potential which is zero inside the nuclear radius and infinite outside that radius.

In spherical polar coordinates, the Shrodinger equation is separable in the general form Y(r,q,f) = R(r)Q(q)F(f), as it is in the case of the hydrogen atom solution. In this case with zero potential, the separation of the azimuthal (f) and colatitude (q) equations requires

The solutions for Q and F, when normalized, give a standard set of functions called spherical harmonics.